Division

Division is a mathematical operation that shows how many times one number (the dividend) can fit into another (the divisor). The answer is called the quotient.

In other words, the quotient (c) is the number that, when multiplied by the divisor (b), gives the dividend (a) as the result: $$ c \times b = a $$

For instance, if you have 12 candies and 3 friends, and you want to share the candies equally, you would calculate:

$$ 12 \div 3 = 4 $$

Each friend gets 4 candies. In this case, 12 is the dividend, 3 is the divisor, and 4 is the quotient.

So, here, 12 is the dividend, 3 is the divisor, and 4 is the quotient.

However, whole number division isn’t always possible because sometimes the dividend isn’t an exact multiple of the divisor. In such cases, you’ll either have a remainder, or if you’re aiming for precision, you’ll end up with a fraction or decimal.

Division with a Remainder

When dividing a number that doesn’t split evenly, there’s a leftover amount called the remainder.

For example, 14 isn’t divisible by 3 exactly.

$$ 14 \div 3 $$

The number 3 doesn’t fit into 14 perfectly. It goes in 4 times (since \(3 \times 4 = 12\)), but there are 2 units left.

So, 14 divided by 3 gives a quotient of 4, with a remainder of 2:

$$ 14 \div 3 = 4 \text{ with a remainder of } 2 $$

This means you can make 4 groups of 3, with 2 left over.

Division with Decimals

If you prefer an exact result without a remainder, you can use decimal numbers.

For example, dividing 17 by 5:

$$ 17 \div 5 $$

Instead of expressing the result with a remainder,

$$ 17 \div 5 = 3 \text{ with a remainder of } 2 $$

you can write it as a decimal:

$$ 17 \div 5 = 3.4 $$

Here, the result is 3.4, a decimal.

This means that 17 has been divided into 5 equal parts, with each part being 3.4 units.

In other words, it’s an exact division expressed in decimals.

How to Perform Long Division

Long division might seem tricky at first, but with a straightforward approach, it becomes a pretty mechanical process.

Here’s how to do long division, which is particularly useful for larger numbers.

Imagine you need to divide 432 by 3. Follow these steps:

1. Write the dividend and divisor
   Write the dividend (432) on the left and the divisor (3) to the right, above the line.
   
2. Divide the first digit
   Start with the first digit of the dividend on the left, which is 4 here. Ask yourself, "How many times does 3 fit into 4?" The answer is 1.
   Write 1 below the division line.
   
3. Multiply and subtract
   Multiply 1 by 3, which equals 3. Write this 3 below the 4 and subtract: \( 4 - 3 = 1 \).
   
4. Bring down the next digit
   Bring down the next digit of the dividend (3) and write it beside the result of the subtraction, forming 13.
   
5. Divide the new number
   Now ask, "How many times does 3 fit into 13?" The answer is 4. Write 4 below the line, next to the previous number.
   
6. Multiply and subtract again
   Multiply 4 by 3, which gives 12. Write 12 below the 13 and subtract: \( 13 - 12 = 1 \).
   
7. Bring down the next digit
   Bring down the next digit of the dividend (2), forming 12.
   
8. Divide the new number
   How many times does 3 fit into 12? It fits exactly 4 times. So, write 4 below the division line, next to the previous numbers.
   
9. Multiply and subtract one last time
   Multiply 4 by 3, which gives 12. Subtract \( 12 - 12 = 0 \). Since there’s no remainder, the division is complete. The result is \( 432 \div 3 = 144 \).
   

This method works for any number and lets you tackle complex divisions systematically.

If the remainder is smaller than the divisor, add a zero to the remainder’s right and place a decimal point in the quotient. Continue until you get the decimal precision you need or until the remainder is zero.

Example

Here’s an example of decimal division with a remainder.

Imagine dividing 25 by 4:

1. Write the long division
   In this case, the dividend is 25, and the divisor is 4.
   
2. Carry out the division
   4 goes into 25 six times because \( 4 \times 6 = 24 \). Write 6 under the line and subtract \( 25 - 24 = 1 \).
   
3. Handle the remainder
   At this point, 1 is less than 4. To proceed, add a zero to the right of 1, making 10. Add a decimal point in the quotient (after the 6) to indicate the decimal part.
   
4. Continue with the decimal
   4 fits into 10 twice because \( 4 \times 2 = 8 \). Add 2 after the decimal in the quotient, making it 6.2. Then subtract \( 10 - 8 = 2 \).
   
5. Repeat the process
   Since 2 is less than 4, add another zero, forming 20.
   
   4 fits into 20 five times, so add 5 to the quotient, making it 6.25. Subtract \( 20 - 20 = 0 \), and there’s nothing left to divide.
   

The division result is

$$ 25 \div 4 = 6.25 $$

In summary:

• If the remainder is smaller than the divisor, add a zero and continue dividing.
• Keep going until you reach the desired decimal precision or until the remainder is zero.

This approach allows you to achieve the necessary decimal precision with ease.

Example (Long Division)

When both the dividend and the divisor have multiple digits, you can make the calculation easier by focusing on their most significant digits.

Here's an example: divide 455 by 63.

At this stage, you could try counting how many times 63 fits into 455, but doing that mentally might be challenging.

To simplify, instead, check how many times 6 fits into 45.

In this case, 6 is the most significant digit of the divisor (63), and 45 is the most significant part of the dividend (455) that can accommodate 6.

Here, 6 fits into 45 seven times, so you write 7 in the quotient.

Next, multiply the divisor, 63, by 7, and write the result, 63x7=441, below the dividend (455). 

Then, subtract 441 from 455, which gives a remainder of 14.

This gives you the outcome of the division: the divisor 63 fits into the dividend 455 a total of 7 times (quotient), leaving a remainder of 14.

If you prefer, you can continue calculating the quotient as we did in the previous example.

Note that this method simplifies the process, but it’s an approximation and doesn’t always yield the correct result. After making an estimate, you need to consider all the digits of the divisor to verify the accuracy, as focusing only on the initial digits can be misleading. For instance, when dividing 935 by 47, you might start by checking how many times 4 (the most significant digit of 47) fits into 9 (the most significant digit of 935). The 4 fits twice into 9. However, if you multiply 2 by the divisor, you get \( 2 \times 47 = 94 \), which is greater than 93. In this scenario, 2 isn’t a valid quotient because the subsequent digits of the divisor influence the calculation.

Division as Fractions

You can express division as a fraction, with the dividend as the numerator and the divisor as the denominator.

For example, if you have 14 candies and 3 friends, and you want to give each an exact portion, you can write the division as either \( 14 \div 3 = 4.67 \) or as a fraction:

$$ \frac{14}{3} = 4.67 $$

So, each friend receives 4.67 candies.

Division as Repeated Subtraction

Another way to understand division: 12 divided by 3 means repeatedly subtracting 3 from 12 until you reach zero:

$$ 12 - 3 - 3 - 3 - 3 = 0 $$

Therefore, 12 divided by 3 is 4 because you subtracted 3 a total of 4 times.

Quotient of Two Signed Numbers

The quotient of two integers has an absolute value equal to the quotient of their absolute values. The sign is positive if the numbers have the same sign and negative if they have opposite signs. If the numerator is zero and the denominator is not zero, the result is zero.

In practical terms, for two integers \( a \) and \( b \), with \( b \neq 0 \), the absolute value of the quotient \( \frac{a}{b} \) will be \(  \frac{|a|}{|b|}  \).

To determine the sign of the quotient, you can apply the same sign rule used for multiplication.

• The quotient is positive (+) if both numbers have the same sign (either both positive or both negative).
• The quotient is negative (-) if the numbers have opposite signs (one positive and one negative).

Remember, two numbers are "like-signed" if they share the same sign. If their signs differ, they are "opposite-signed." For example, 2 and 3 are like-signed, as are -2 and -3, whereas 2 and -3 are opposite-signed.

Example

Let's look at the following division:

$$ 8 \div 2 $$

Here, the dividend \( a = 8 \) and the divisor \( b = 2 \) are both positive, so the quotient will be positive as well.

$$ 8 \div 2 = 4 $$

In every case, calculate the quotient by dividing the absolute value of the dividend by the absolute value of the divisor. The sign depends on whether the numbers have the same sign. $$ 8 \div 2 = + (|8| \div |2|) = +4 $$

Similarly, if both the dividend and divisor are negative, the result will also be positive.

$$ (-8) \div (-2) = 4 $$

Since both numbers are negative, they are like-signed, and the quotient is positive. $$ (-8) \div (-2) = +(|-8| \div |-2|) = +(8 \div 2) = +4 $$

If the signs differ, however, the quotient will be negative because the numbers are opposite-signed.

$$ (-8) \div  2 = -4 $$

$$ 8 \div (-2) = -4 $$

In both cases, the numbers have opposite signs, so the quotient is negative. $$ (-8) \div 2 = -(|-8| \div |2|) = -(8 \div 2) = -4 $$ $$ 8 \div (-2) = -(|8| \div |-2|) = -(8 \div 2) = -4 $$

Lastly, if the numerator is zero (i.e., \( a = 0 \)) and the denominator \( b \) is not zero, the result is always zero.

$$ 0 \div 2 = 0 $$

$$ 0 \div (-2) = 0 $$

By following these straightforward rules, you can calculate the quotient for any pair of signed integers.

The Quotient of Rational Numbers

The quotient of two rational numbers, as long as the second number is not zero, is equal to the product of the first number and the reciprocal of the second. $$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} $$

Rational numbers are numbers that can be written as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).

For example, \( \frac{3}{4} \), \( -\frac{2}{5} \), and \( 7 \) (which can be expressed as \( \frac{7}{1} \)) are all rational numbers.

To find the quotient of two rational numbers, \( \frac{a}{b} \) and \( \frac{c}{d} \) (with \( c \neq 0 \) to avoid dividing by zero), you can think of the operation as:

$$ \frac{\frac{a}{b}}{\frac{c}{d}} $$

Dividing fractions might seem tricky, but it’s actually quite simple thanks to a key property: dividing by a fraction is the same as multiplying by its reciprocal.

$$ \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \cdot \frac{d}{c} = \frac{a \cdot d}{b \cdot c} $$

Here, the reciprocal of a fraction is obtained by swapping the numerator and the denominator. For instance, the reciprocal of \( \frac{c}{d} \) is \( \frac{d}{c} \).

In essence, division can be understood as the reverse of multiplication, with the reciprocal playing a crucial role. This approach makes it easier to work with fractions and simplifies otherwise complex operations.

A Practical Example

Let’s calculate the following division:

$$ \frac{3}{4} \div \frac{2}{5} $$

First, rewrite the division as a ratio of two fractions:

$$ \frac{\frac{3}{4}}{\frac{2}{5}} $$

Next, apply the rule by multiplying the first fraction by the reciprocal of the second:

$$ \frac{\frac{3}{4}}{\frac{2}{5}} = \frac{3}{4} \cdot \frac{5}{2} $$

Now multiply the numerators and denominators:

$$ \frac{3 \cdot 5}{4 \cdot 2} = \frac{15}{8} $$

So, the result of the division is \( \frac{15}{8} \).

Division Properties

Division has unique properties that distinguish it from other operations, like addition or multiplication. Let’s explore these:

• Invariant Property of Division
  When you multiply or divide both the dividend and the divisor by the same nonzero number \( k \), the quotient remains the same. $$ \frac{a}{b} = \frac{a \cdot k}{b \cdot k} = \frac{a \div k}{b \div k} $$ Here, \( a \) represents the dividend, \( b \) is the divisor, and \( k \) is any number other than zero.

  Example: In the division $ 8 \div 4 $, if you multiply or divide both the dividend and the divisor by two, the quotient stays the same.  $$  8 \div 4 = (8 \times 2) \div (4 \times 2) = (8 \div 2) \div (4 \div 2) = 2 $$ It's important that $ k $ is never zero to avoid a division by zero.

• Distributive Property of Division Only Applies to the Right
  The distributive property of division over addition and subtraction only applies when the divisor is distributed to the sum or difference on the right side. Furthermore, the divisor must always be non-zero. $$ (a + b) ÷ c = a ÷ c + b ÷ c $$ $$ (a - b) ÷ c = a ÷ c - b ÷ c $$

  Example: Consider this expression $$ (8+4) ÷ 2  $$ You can reach the same result by distributing the divisor to each term  $$ (8+4) ÷ 2 = (8 ÷ 2) + (4 ÷ 2)  $$ When you carry out the calculations, both sides of the equation yield the same result $$ (8+4) ÷ 2 = (8 ÷ 2) + (4 ÷ 2)  $$ $$ 12 ÷ 2 = 4 + 2  $$ $$ 6 = 6 $$

• Non-commutative
  Commutativity means you can swap elements without changing the result. This is true for addition and multiplication (e.g., \( 2 + 3 = 3 + 2 \)), but not for division.

  Example: $$ 8 \div 4 = 2 $$ but
  $$ 4 \div 8 = 0.5 $$ Thus, the order matters in division, and switching it changes the result.

• Non-associative
  Associativity allows you to change the grouping of numbers without affecting the result, as in addition \((a + b) + c = a + (b + c)\). Division, however, is not associative.

  Example: $$ (24 \div 4) \div 2 = 6 \div 2 = 3 $$ but $$ 24 \div (4 \div 2) = 24 \div 2 = 12 $$ Here, changing the way numbers are grouped leads to different results, so it’s essential to follow the order of operations.

• Identity element
  In division, the identity element is 1. When dividing a number by 1, the result is always the original number. Example: $$ 9 \div 1 = 9 $$ This is because any number divided by 1 remains unchanged.
• Division by zero is undefined
  Dividing any number by zero is impossible. Division by zero is undefined and would lead to nonsensical and undefined results.

  Example: We can’t calculate: $$ 8 \div 0 $$

• Division of zero
  However, when zero is divided by another number (other than zero), the result is always zero.

  Example: $$ 0 \div 5 = 0 $$ Essentially, dividing zero by any number always yields zero.

• Division is Closed for Rational Numbers
  This means that the division of any two rational numbers will always produce another rational number.

  Example: Consider two rational numbers \( \frac{2}{3} \) and \( \frac{5}{7} \). Their division is: $$ \frac{2}{3} \div \frac{5}{7} = \frac{2}{3} \cdot \frac{7}{5} = \frac{2 \cdot 7}{3 \cdot 5} = \frac{14}{15} $$ The result, \( \frac{14}{15} \), is also a rational number.

• Division is not a closed operation within natural numbers or integers
  Division is not a closed operation in the set of natural numbers \( \mathbb{N} \), as the result of dividing two natural numbers is not always a natural number. For the same reason, it’s not a closed operation within the set of integers either.

  Example: The division \( 5 \div 2 = 2.5 \) does not belong to the set of natural numbers \( \mathbb{N} \), since natural numbers are whole and non-negative. In this case, the quotient of the two natural numbers (5 and 2) is a rational number (2.5) $$ 5 \div 2 = 2.5 \in \mathbb{Q} $$ So, the result is not a natural number $ 2.5 \notin \mathbb{N} $.